Agriculture played a large role in many parts of the world, but did it play a role in the far north as well, where the weather was very cold and much of the land was tundra? Or did people rely solely on herding and hunting/gathering? If it existed, how would this agriculture differ from what existed in more temperate regions?
What do you mean by "far north," exactly? I'm assuming Arctic Circle but that may not be what you meant.
Anyway, what you're really asking about is arable land. And thanks to all the changes surrounding the agricultural industry in the Soviet Union and its successor states, we have lots of information on the agricultural industries in those countries. And we have maps like this, showing us the arable soil in those countries.
And arable soil is ultimately what you're asking about. Arable literally means "plow-able", with the implication being that you can actually work arable soil (amusingly, a very literal and utterly unhelpful interpretation of the etymology of the term means that glaciers are theoretically arable). But I digress.
If you don't have arable soil, you can't have agriculture. There are some ways around this. In the case of Israel, for instance, you can see the value of attempts to reverse desertification. You can circumvent the need for arable soil by using hydroponics and related techniques. The ISS has recently started growing its own food with aeroponics. But as cool as these things are (c'mon, space zucchini?), they're not really techniques that you can use to circumvent the need for arable soil on a large scale. Yet.
Which brings us back to your question and this map of Soviet land use. If you look closely at that map, you can see little isolated pockets of dark green land. One of the furthest north of those is in the general vicinity of Kandalaksha, which is on the southwest corner of the Kola peninsula, situated at around 67 degrees north. The Arctic Circle begins around 66 degrees north. You can see a few isolated pockets in northern Siberia that are also north of the Circle. So, yeah, farming in the far north definitely happened. It just wasn't very large scale.
If you lived far away from where you could realistically subsist from farming (either your own or someone else's), your diet was likely heavily on hunting of some kind. Seal, walrus, caribou, birds, fish -- that kind of thing. You'd augment this diet with tubers (usually distant cousins of the potato, like this sample of mousefood) and wild berries. But you'd be eating a lot of salmon or a lot of reindeer, depending on where you lived.
